Almost Quantum Correlations are Inconsistent with Specker's Principle

Ernst Specker considered a particular feature of quantum theory to be especially fundamental, namely that pairwise joint measurability implies global joint measurability for sharp measurements [vimeo.com/52923835 (2009)]. To date, it seemed that Speckeru0027s principle failed to single out quantum theory from the space of all general probabilistic theories. In particular, consistent exclusivity --- an important consequence of Speckeru0027s principle --- is satisfied by both quantum and almost quantum correlations. Here, we identify another statistical implication of Speckeru0027s principle besides consistent exclusivity, which possibly holds for almost quantum correlations. However, our main result asserts that Speckeru0027s principle cannot be satisfied in any theory that yields almost quantum models.